A contribution of the current paper is to show how the resolution principle [1]}
can be used in handling degree-4 variables in solving the MaxSAT problem.
It has been well-known that the resolution principle is a very powerful tool in solving
the SAT problem [1]}. In particular, variable resolutions in a CNF formula
preserve the satisfiability of the formula. Unfortunately, variable resolutions cannot
be used directly in solving the MaxSAT problem in general case because they
do not provide a predictable decreasing in the maximum number of clauses in the
CNF formulas that can be satisfied by an assignment. In particular, for a degree-4
variable \(x\)  in a CNF formula \(F\)  for which an optimal assignment satisfies \(k\)  clauses,
the resolution on \(x\)  may result in CNF formulas for which optimal assignments
satisfy \(k-4\) , \(k-2\) , \(k-1\) , and \(k\)  clauses, respectively.
